A new upper bound on the specific free energy of dilute Bose gases

We prove an upper bound for the free energy (per unit volume) of the dilute Bose gas in the thermodynamic limit, showing that the free energy at density $\rho$ and inverse temperature $\beta$ differs from that of the non-interacting system by the correction term $4\pi \mathfrak{a}(2{\rho }^2-[\rho -{\rho }_{\mathsf{\text{c}}}(\beta ){]}_{+}^2)$. Here, $\mathfrak{a}$ denotes the scattering length of the interaction potential, ${\rho }_{\mathsf{\text{c}}}(\beta )$ the critical density for Bose-Einstein condensation of the non-interacting gas and $[\cdot {]}_{+}=\max \{0,\cdot \}$. This result was previously established by Yin in [37]. Our proof applies to a broader class of interaction potentials, yields a better rate, and we believe it has potential for further extensions.